Two-Loop Superstrings II, The Chiral Measure on Moduli Space

TL;DR

This work resolves long-standing ambiguities in the two-loop chiral measure for superstrings by introducing a gauge-fixing scheme based on projecting supergeometries onto the supersymmetric super period matrix $\hat{\Omega}_{IJ}$. The authors perform a careful chiral splitting, parametrize supermoduli as a fiber over $\hat{\Omega}_{IJ}$, and handle the odd moduli with a finite-dimensional superdeterminant that is rendered slice-independent by stress-tensor insertions and Beltrami variations. In genus 2, they derive an explicit gauge-fixed chiral measure expressed through determinants and correlators, and prove its independence under infinitesimal changes of both the diffeomorphism and supersymmetry slices. The resulting formulation yields an unambiguous, fully explicit description of the genus-2 chiral measure in terms of modular forms and holomorphic data, paving the way for exact two-loop superstring amplitudes. This advances the program of obtaining a completely explicit, gauge-invariant framework for string perturbation theory beyond the first loop.

Abstract

A detailed derivation from first principles is given for the unambiguous and slice-independent formula for the two-loop superstring chiral measure which was announced in the first paper of this series. Supergeometries are projected onto their super period matrices, and the integration over odd supermoduli is performed by integrating over the fibers of this projection. The subtleties associated with this procedure are identified. They require the inclusion of some new finite-dimensional Jacobian superdeterminants, a deformation of the worldsheet correlation functions using the stress tensor, and perhaps paradoxically, another additional gauge choice, ``slice \hatμchoice'', whose independence also has to be established. This is done using an important correspondence between superholomorphic notions with respect to a supergeometry and holomorphic notions with respect to its super period matrix. Altogether, the subtleties produce precisely the corrective terms which restore the independence of the resulting gauge-fixed formula under infinitesimal changes of gauge-slice. This independence is a key criterion for any gauge-fixed formula and hence is verified in detail.